S.O.S. Mathematics CyberBoard

Your Resource for mathematics help on the web!
It is currently Sun, 26 May 2013 04:16:51 UTC

View unanswered posts | View active topics


Board index » High School and College Mathematics » Matrix Algebra

All times are UTC [ DST ]



Post new topic Reply to topic  Page 1 of 1
  [ 8 posts ] 
  Print view Previous topic | Next topic 
Author Message
TsAmE
 Post subject: Matrix multiplication problem
PostPosted: Fri, 8 Oct 2010 22:45:25 UTC 
Offline
Senior Member

Joined: Mon, 19 Apr 2010 13:45:05 UTC
Posts: 61
Find all matrices X satisfying AX = XA, where A = \begin{pmatrix}1 & 0\\ 2& 0\end{pmatrix}.

\begin{pmatrix}1 & 0\\ 2& 0\end{pmatrix}\begin{pmatrix}x1 & y1\\ x2& y2\end{pmatrix} = \begin{pmatrix}x1 & y1\\ x2& y2\end{pmatrix}\begin{pmatrix}1 & 0\\ 2& 0\end{pmatrix}
\begin{pmatrix}x1 & y1\\ 2x1& 2y1\end{pmatrix}=\begin{pmatrix}x1 & 2y1\\ x2& 2y2\end{pmatrix}

I dont know what more to do


Last edited by TsAmE on Sat, 9 Oct 2010 14:53:19 UTC, edited 5 times in total.

Top
 Profile  
 
Matt
 Post subject:
PostPosted: Fri, 8 Oct 2010 23:00:00 UTC 
Offline
Member of the 'S.O.S. Math' Hall of Fame
User avatar

Joined: Wed, 1 Oct 2003 04:45:43 UTC
Posts: 9633
Double check your work. The matrices weren't multiplied correctly.
Top
 Profile  
 
TsAmE
 Post subject:
PostPosted: Sat, 9 Oct 2010 13:46:10 UTC 
Offline
Senior Member

Joined: Mon, 19 Apr 2010 13:45:05 UTC
Posts: 61
I forgot the equals sign for the last step. I have edited the post. Otherwise I check my multiplication and didnt see anything wrong.
Top
 Profile  
 
outermeasure
 Post subject:
PostPosted: Sat, 9 Oct 2010 13:57:48 UTC 
Online
Moderator
User avatar

Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6010
Location: 127.0.0.1, ::1 (avatar courtesy of UDN)
TsAmE wrote:
I forgot the equals sign for the last step. I have edited the post. Otherwise I check my multiplication and didnt see anything wrong.


Are you sure you copied A correctly?

TsAmE wrote:
Find all matrices X satisfying AX = XA, where A = \begin{pmatrix}1 & 2\\ 2& 0\end{pmatrix}.

\begin{pmatrix}1 & 0\\ 2& 0\end{pmatrix}\begin{pmatrix}x1 & y1\\ x2& y2\end{pmatrix} = \begin{pmatrix}x1 & y1\\ x2& y2\end{pmatrix}\begin{pmatrix}1 & 0\\ 2& 0\end{pmatrix}
\begin{pmatrix}x1 & y1\\ 2x1& 2y1\end{pmatrix}=\begin{pmatrix}x1 & 2y1\\ x2& 2y2\end{pmatrix}

I dont know what more to do

_________________
\begin{aligned} Spin(1)&=O(1)=\mathbb{Z}/2&\quad&\text{and}\\ Spin(2)&=U(1)=SO(2)&&\text{are obvious}\\ Spin(3)&=Sp(1)=SU(2)&&\text{by }q\mapsto(\mathop{\mathrm{Im}}\mathbb{H}\ni p\mapsto qp\bar{q})\\ Spin(4)&=Sp(1)\times Sp(1)&&\text{by }(q_1,q_2)\mapsto(\mathbb{H}\ni p\mapsto q_1p\bar{q_2})\\ Spin(5)&=Sp(2)&&\text{by }\mathbb{HP}^1\cong S^4_{round}\hookrightarrow\mathbb{R}^5\\ Spin(6)&=SU(4)&&\text{by the irrep }\Lambda_+\mathbb{C}^4 \end{aligned}
Top
 Profile  
 
TsAmE
 Post subject:
PostPosted: Sat, 9 Oct 2010 14:56:12 UTC 
Offline
Senior Member

Joined: Mon, 19 Apr 2010 13:45:05 UTC
Posts: 61
Sorry made a mistake A is suppose to = \begin{pmatrix}1 & 0\\ 2& 0\end{pmatrix}.

I found my mistake. Here is my correction:

\begin{pmatrix}x1 &y1 \\ 2x1& 2y1\end{pmatrix} = \begin{pmatrix}x1 + 2y1 &0 \\ x2 + 2y2 & 0\end{pmatrix}

Im not sure what to do from here.
Top
 Profile  
 
helmut
 Post subject:
PostPosted: Sat, 9 Oct 2010 16:21:50 UTC 
Offline
Site Admin
User avatar

Joined: Sat, 26 Apr 2003 22:14:40 UTC
Posts: 2063
Location: El Paso TX (USA)
Two matrices are equal if their respective entries are equal. That gives you four equations...

_________________
The greater danger for most of us lies not in setting our aim too high and falling short; but in setting our aim too low, and achieving our mark. - Michelangelo Buonarroti
Top
 Profile  
 
TsAmE
 Post subject:
PostPosted: Sat, 9 Oct 2010 19:20:50 UTC 
Offline
Senior Member

Joined: Mon, 19 Apr 2010 13:45:05 UTC
Posts: 61
helmut wrote:
Two matrices are equal if their respective entries are equal. That gives you four equations...


I then get the equations:

x1 = x1 + 2y1
y1 = 0
2x1 = x2 + 2y2
2y1 = 0

From this info I found that:

y1 = 0
x1 = x1
Cant solve for x2 :|
Top
 Profile  
 
Soroban
 Post subject: Re: Matrix multiplication problem
PostPosted: Sun, 10 Oct 2010 15:02:47 UTC 
Offline
Member of the 'S.O.S. Math' Hall of Fame

Joined: Mon, 19 May 2003 19:55:19 UTC
Posts: 7952
Location: Lexington, MA
Hello, TsAmE!

Quote:
Find all matrices X satisfying AX = XA, where A \:=\:\begin{pmatrix}1 & 0\\ 2& 0\end{pmatrix}.

Let: .X \:=\:\begin{pmatrix}a&b\\c&d\end{pmatrix}

Then we have: .\begin{pmatrix}1&0\\2&0\end{pmatrix} \begin{pmatrix}a&b\\c&d\end{pmatrix} \;=\; \begin{pmatrix}a&b\\c&d\end{pmatrix}\begin{pmatrix}1&0\\2&0\end{pmatrix}

. . . . . . . . . . . . . . . . \begin{pmatrix}a & b \\ 2a & 2b\end{pmatrix} \;=\;\begin{pmatrix}a+2b & 0 \\ c + 2d & 0 \end{pmatrix}


Hence, we have four equations: . \begin{array}{cccccccccc}a &=& a+2b & [1] && b&=& 0 & [2] \\ 2a &=& c+2d & [3] && 2b &=& 0 & [4] \end{array}


From [3] and [4] we have: .b \:=\: 0

. . . . . Then [1] becomes: .a \:=\: a

. . .And from [3] we have: .d \:=\:\dfrac{2a-c}{2}


Therefore: .X \;=\; \begin{pmatrix}a & 0 \\ c & \frac{2a-c}{2} \end{pmatrix} . for any values of a and c.
Top
 Profile  
 
Display posts from previous:  Sort by  
Post new topic Reply to topic  Page 1 of 1
  [ 8 posts ] 

Board index » High School and College Mathematics » Matrix Algebra

All times are UTC [ DST ]

Who is online

Users browsing this forum: No registered users

You cannot post new topics in this forum
You cannot reply to topics in this forum
You cannot edit your posts in this forum
You cannot delete your posts in this forum

Search for:
Jump to:  
Contact Us | S.O.S. Mathematics Homepage
Privacy Statement | Search the "old" CyberBoard

users online during the last hour
Powered by phpBB © 2001, 2005-2011 phpBB Group.
Copyright © 1999-2013 MathMedics, LLC. All rights reserved.
Math Medics, LLC. - P.O. Box 12395 - El Paso TX 79913 - USA